cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A124258 Triangle whose rows are sequences of increasing and decreasing squares: 1; 1,4,1; 1,4,9,4,1; ...

Original entry on oeis.org

1, 1, 4, 1, 1, 4, 9, 4, 1, 1, 4, 9, 16, 9, 4, 1, 1, 4, 9, 16, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 49, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 49, 64, 49, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 49, 64, 81, 64, 49, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16
Offset: 1

Views

Author

Jonathan Vos Post, Dec 16 2006

Keywords

Comments

The triangle A003983 with individual entries squared and each 2nd row skipped.
Analogous to A004737. - Peter Bala, Sep 25 2007
T(n,k) = min(n,k)^2. The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013

Examples

			Triangle starts
  1;
  1, 4, 1;
  1, 4, 9, 4, 1:
  1, 4, 9, 16, 9, 4, 1:
From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as table:
  1...1...1...1...1...1...
  1...4...4...4...4...4...
  1...4...9...9...9...9...
  1...4...9..16..16..16...
  1...4...9..16..25..25...
  1...4...9..16..25..36...
  ...
The start of the sequence as triangle array read by rows:
  1;
  1, 4, 1;
  1, 4, 9,  4,  1;
  1, 4, 9, 16,  9,  4,  1;
  1, 4, 9, 16, 25, 16,  9,  4, 1;
  1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 1;
  ...
Row number k contains 2*k-1 numbers 1,4,...,(k-1)^2,k^2,(k-1)^2,...,4,1. (End)

Links

Crossrefs

Cf. A005900 (row sums), A004737, A133819, A133823, A133824, A133825, A003983.

Programs

  • Maple
    A003983 := proc(n,k) min(n,k) ; end: A124258 := proc(n,k) A003983(n,k)^2 ; end: for d from 1 to 20 by 2 do for c from 1 to d do printf("%d, ",A124258(d+1-c,c)) ; od: od: # R. J. Mathar, Sep 21 2007
# second Maple program:
T:= n-> i^2$i=1..n, (n-i)^2$i=1..n-1:
seq(T(n), n=1..10);  # Alois P. Heinz, Feb 15 2022
  • Mathematica
    Flatten[Table[Join[Range[n]^2,Range[n-1,1,-1]^2],{n,10}]] (* Harvey P. Dale, Jun 14 2015 *)

Formula

O.g.f.: (1+qx)^2/((1-x)(1-qx)^2(1-q^2x)) = 1 + x(1 + 4q + q^2) + x^2(1 + 4q + 9q^2 + 4q^3 + q^4) + ... . - Peter Bala, Sep 25 2007
From Boris Putievskiy, Jan 13 2013: (Start)
a(n) = (A004737(n))^2.
a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^2. (End)

Extensions

More terms from R. J. Mathar, Sep 21 2007
Edited by N. J. A. Sloane, Jun 30 at the suggestion of R. J. Mathar

A133824 Triangle whose rows are sequences of increasing and decreasing fourth powers: 1; 1,16,1; 1,16,81,16,1; ... .

Original entry on oeis.org

1, 1, 16, 1, 1, 16, 81, 16, 1, 1, 16, 81, 256, 81, 16, 1, 1, 16, 81, 256, 625, 256, 81, 16, 1, 1, 16, 81, 256, 625, 1296, 625, 256, 81, 16, 1, 1, 16, 81, 256, 625, 1296, 2401, 1296, 625, 256, 81, 16, 1, 1, 16, 81, 256, 625, 1296, 2401, 4096, 2401, 1296, 625, 256, 81, 16
Offset: 0

Views

Author

Peter Bala, Sep 25 2007

Keywords

Comments

Reading the triangle by rows produces the sequence 1,1,16,1,1,16,81,16,1,..., analogous to A004737.
From - Boris Putievskiy, Jan 13 2013: (Start)
The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1).
Row number k contains 2*k-1 numbers 1,16,...,(k-1)^4,k^4,(k-1)^4,...,16,1. (End)

Examples

			Triangle starts:
  1;
  1, 16, 1;
  1, 16, 81, 16, 1;
  1, 16, 81, 256, 81, 16, 1;
  ...
From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as table:
  1...1...1...1...1.. .1...
  1..16..16..16..16...16...
  1..16..81..81..81...81...
  1..16..81.256.256..256...
  1..16..81.256.625..625...
  1..16..81.256.625.1296...
  ...
(End)

Links

Crossrefs

Cf. A004737, A061803 (row sums), A133821, A124258, A133823, A003983.

Programs

  • Mathematica
    p4[n_]:=Module[{c=Range[n]^4},Join[c,Rest[Reverse[c]]]]; Flatten[p4/@ Range[10]] (* Harvey P. Dale, Dec 08 2014 *)

Formula

O.g.f.: (1+qx)(1+11qx+11q^2x^2+q^3x^3)/((1-x)(1-qx)^4(1-q^2x)) = 1 + x(1 + 16q + q^2) + x^2(1 + 16q + 81q^2 + 16q^3 + q^4) + ... . Cf. 4th row of A008292.
From Boris Putievskiy, Jan 13 2013: (Start)
T(n,k) = min(n,k)^4.
a(n) = (A004737(n))^4.
a(n) = (A124258(n))^2.
a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^4. (End)

A133820 Triangle whose rows are sequences of increasing cubes: 1; 1,8; 1,8,27; ... .

Original entry on oeis.org

1, 1, 8, 1, 8, 27, 1, 8, 27, 64, 1, 8, 27, 64, 125, 1, 8, 27, 64, 125, 216, 1, 8, 27, 64, 125, 216, 343, 1, 8, 27, 64, 125, 216, 343, 512, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Offset: 1

Views

Author

Peter Bala, Sep 25 2007

Keywords

Comments

Reading the triangle by rows produces the sequence 1,1,8,1,8,27,1,8,27,64,..., analogous to A002260.

Examples

			Triangle starts
1;
1, 8;
1, 8, 27;
1, 8, 27, 64;
1, 8, 27, 64, 125;

Links

Crossrefs

Cf. A000537 (row sums), A002260, A019522, A133819, A133821, A133823.

Programs

  • Haskell
    a133820 n k = a133820_tabl !! (n-1) !! (k-1)
a133820_row n = a133820_tabl !! (n-1)
a133820_tabl = map (`take` (tail a000578_list)) [1..]
-- Reinhard Zumkeller, Nov 11 2012
  • Mathematica
    Module[{nn=10,c},c=Range[nn]^3;Flatten[Table[Take[c,n],{n,10}]]] (* Harvey P. Dale, Mar 05 2014 *)

Formula

O.g.f.: (1+4qx+q^2x^2)/((1-x)(1-qx)^4) = 1 + x(1 + 8q) + x^2(1 + 8q + 27q^2) + ... .

Extensions

Offset changed by Reinhard Zumkeller, Nov 11 2012
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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